In order to bound \({T_{i,n}}\) , we can split \({T_{i,n}}\)  into two parts using a key lemma by [1]} to yield \({T_{i,n}} \le \Lambda _1 + \Lambda _2 +1\) , where \(\Lambda _1 = {\sum ^{n-1}_{s=0}{\frac{1}{G_{1,s}}-1}}\)  and \(\Lambda _2 = \sum ^{n-1}_{s=0}\mathbb {P}{G_{1,s} > \frac{1}{n}}\) . It remains to upper bound \(\Lambda _1\)  and \(\Lambda _2\) . The techniques to upper bound \(\Lambda _1\)  are similar to those from [2]} and [3]}. To upper bound \(\Lambda _2\) , we split the event \(E_i^{c}(t) = {\hat{}_{\gamma }(i,t) \le _{\gamma }(1) + }\)  into

1() = -i,t +i -(ER(i,) - ),
